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Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech

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Building short walls How many ways are there to build a 2 x n wall with 1 x 2 bricks? 2 n 2 1

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Building short walls 2 1How many ways are there to build a 2 x n wall with 1 x 2 bricks? 2 n

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n=1 : n=2 : n=3 : n=0 : n=4 : Building short walls

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n=0 : n=1 : n=2 : n=3 : n=4 : The number of walls equal: f n = 1, 1, 2, 3, 5

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Building short walls n=0 : n=1 : n=2 : n=3 : n=4 : The number of walls equal: f n = 1, 1, 2, 3, 5, 8, 13, 21,... ?

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Building short walls n=0 : n=1 : n=2 : n=3 : n=4 : The number of walls equals: f n = 1, 1, 2, 3, 5, 8, 13, 21,... ? n 2 2 fn =fn = {

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Building short walls n=0 : n=1 : n=2 : n=3 : n=4 : The number of walls equals: f n = 1, 1, 2, 3, 5, 8, 13, 21,... ? n 2 2 fn =fn = {

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Building short walls n=0 : n=1 : n=2 : n=3 : n=4 : The number of walls equals: f n = 1, 1, 2, 3, 5, 8, 13, 21,... ? n 2 2 f n-1 f n-2 fn =fn = {

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The Fibonacci Numbers The number of walls equals: f n = 1, 1, 2, 3, 5, 8, 13, 21,... f n-1 f n-2 fn =fn = { f n = f n-1 + f n-2, f 0 = f 1 = 1 f n = (φ n + (1-φ) n ) /√5, where: (“golden ratio”) φ = 1+√5 2

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Domino Tilings Given a region R on the infinite chessboard, cover with non-overlapping 2 x 1 dominos. Where is a tiling? Do any even exist? How many tilings are there? What does a typical tiling look like? When do we stop our algorithms? Why do we care?

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Where is a tiling? Do any exist? n n ★ Only if n is even!

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Where is a tiling? Do any exist? The Area of R must be even n n

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Where is a tiling? Do any exist? n n ★ There must be an equal number of black and white squares. The Area of R must be even

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Where is a tiling? Do any exist? Is this enough? The Area of R must be even With an equal number of white and black squares

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Where is a tiling? Do any exist? Is this enough? ?... There is an efficient algorithm to decide if R is tileable and to find one if it is. [Thurston] The Area of R must be even With an equal number of white and black squares

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Domino Tilings Where is a tiling? Do any even exist? How many tilings are there? What does a typical tiling look like? When do we stop our algorithms? Why do we care?

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How many tilings are there? φ n Short 2 x n walls

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How many tilings are there? φ n Short 2 x n walls Aztec Diamonds n # = 1 2 8 64 n=0 n=1 n=2 n=3 n(n+1)/2 [Elkies, Kuperberg, Larson, Propp]

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How many tilings are there? φ n Short 2 x n walls Aztec Diamonds n n(n+1)/2

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How many tilings are there? φ n Short 2 x n walls Aztec Diamonds Square n x n walls n ? n(n+1)/2

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How many tilings are there? Square n x n walls 2 (Area/4) < # < # < 4 (Area) <

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How many tilings are there? # φ n Short 2 x n walls Aztec Diamonds Square n x n walls 2 Area/4 < # < 4 Area # n(n+1)/2

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How many: An Algorithm Mark alternating vertical edges;

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How many: An Algorithm Mark alternating vertical edges; Use marked tiles; Markings must line up!

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How many: An Algorithm t1t1 t2t2 t3t3 s1s1 s2s2 s3s3

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t1t1 t2t2 t3t3 s1s1 s2s2 s3s3 s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 We want to count non-intersecting sets of paths from s i to t i. [R.]

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How many: An Algorithm We want to count non-intersecting sets of paths from s i to t i. Let a ij be the number of paths from s i to t j. s1s1 s2s2 s3s3 t1t1 t2t2 t3t3

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How many: An Algorithm s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 Let a ij be the number of paths from s i to t j. We want to count non-intersecting sets of paths from s i to t i. 1 4 16 1 3 12 52 1 5 24 1 7 40 1 9 1 10 52 40 10

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How many: An Algorithm s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 We want to count non-intersecting sets of paths from s i to t i. 1 8 1 7 40 1 5 25 0 3 13 62 1 5 24 1 6 30 52 40 10 40 62 30 Let a ij be the number of paths from s i to t j.

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How many: An Algorithm s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 We want to count non-intersecting sets of paths from s i to t i. 1 1 10 1 8 1 6 30 1 4 16 0 2 6 22 52 40 10 40 62 30 10 30 22 Let a ij be the number of paths from s i to t j.

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How many: An Algorithm s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 We want to count non-intersecting sets of paths from s i to t i. 52 40 10 40 62 30 10 30 22 = 1728. [Gessel, Viennot] Let a ij be the number of paths from s i to t j. Det This is the number domino tilings!!

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Proof sketch for two paths: a 11 x a 22 counts what we want + extra stuff. a 12 x a 21 also counts the extra stuff. Therefore (a 11 x a 22 ) - (a 12 x a 21 ) counts real tilings. = # (This is the 2 x 2 determinant!)

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Domino Tilings Where is a tiling? Do any even exist? How many tilings are there? What does a typical tiling look like? When do we stop our algorithms? Why do we care?

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Why Mathematicians Care Arctic Circle Theorem [Jockush, Propp, Shor] The Aztec Diamond

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On the chessboard On the hexagonal lat. “Domino tilings” “Lozenge tilings” (little “cubes”) What about tilings on lattices?

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Why Mathematicians Care

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Why do we care? Mathematics: Discover patterns Chemistry, Biology: Estimate probabilities Physics: Count and calculate other functions to study a physical system Nanotechnology: Model growth processes

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Why Physicists Care “Dimer models”: diatomic molecules adhering to the surface of a crystal. The count (“partition function”) determines: specific heat, entropy, free energy, … What does “nature” compute?

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Domino Tilings Where is a tiling? Do any even exist? How many tilings are there? What does a typical tiling look like? When do we stop our algorithms? Why do we care?

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What does a typical tiling look like?

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“Mix” them up!

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Markov chain for Lozenge Tilings Repeat : Pick v in the lattice region; Add / remove the ``cube’’. at v w.p. ½, if possible. v v

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Markov chain for Lozenge Tilings v 1. The state space is connected. 2.If we do this long enough, each tiling will be equally likely. 3. How long is “long enough” ? v

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Domino Tilings Where is a tiling? Do any even exist? How many tilings are there? What does a typical tiling look like? When do we stop our algorithms? Why do we care?

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When do we stop our algorithms? v v Thm: The lozenge Markov chain is “rapidly mixing.” [Luby, R., Sinclair] 2 n, 3 n n 2, 10 √n, … (exponential) n 2, n log n, n 10, … (polynomial) 3. How long is “long enough” ?

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a 2 x 2 square;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a 2 x 2 square; Rotate, if possible;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a 2 x 2 square; Rotate, if possible; Otherwise do nothing.

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx and a color;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx and a color; Recolor, if possible;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx and a color; Recolor, if possible; Otherwise do nothing.

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx v and a bit b;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx v and a bit b; If b=1, try to add v

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx v and a bit b; If b=1, try to add v; If b=0, try to remove v;

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Pick a vtx v and a bit b; If b=1, try to add v; If b=0, try to remove v; O.w. do nothing.

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What about other models? Potts model Hardcore model Dimer model Domino tilings k-colorings Independent sets Thm: All of these chains are rapidly mixing.

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HOWEVER... Three-colorings: The local chain is fast for 3-colorings in 2-d [LRS] but slow for 3-colorings in sufficiently high dimension. [Galvin, Kahn, R, Sorkin], [Galvin, R] Independent Sets The local chain is fast for sparse Ind Sets in 2-d [Luby, Vigoda],…, [Weitz] but slow for dense Ind Sets. [R.]

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Dense Weighted Independent Sets Sparse Fast Phase Transition Slow

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Why? n 2 /2 (n 2 /2-n/2) “Even” “Odd” #R/#B

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Summary Where How What When Why

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Summary Where How What When Why Algebra Algorithms Combinatorics

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Summary Where How What When Why Algebra Algorithms Combinatorics Geometry Probability Physics

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Summary Where How What When Why Algebra Algorithms Combinatorics Geometry Probability Physics Chemistry Nanotechnology Biology

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THANK YOU ! Math

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THANK YOU ! Math+ CS

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THANK YOU ! Math+ CS + Biology, Chemistry,

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THANK YOU ! + Physics, Nanotechnology,... Math+ CS + Biology, Chemistry,

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THANK YOU !

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